https://file.aiquickdraw.com/mathbot/file/1727103378636lx1kpron.jpg

To expand \( f(x) = (7x - 2)^3 \), we can use the binomial expansion formula, which states:

\[
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
\]

Here, \( a = 7x \) and \( b = -2 \). Let's expand this step-by-step:

1. \( a^3 = (7x)^3 = 343x^3 \)
2. \( 3a^2b = 3(7x)^2(-2) = 3(49x^2)(-2) = -294x^2 \)
3. \( 3ab^2 = 3(7x)(-2)^2 = 3(7x)(4) = 84x \)
4. \( b^3 = (-2)^3 = -8 \)

Thus, the expansion of \( (7x - 2)^3 \) is:

\[
f(x) = 343x^3 - 294x^2 + 84x - 8
\]

Now, the blanks in the provided image would be filled as follows:

- First blank (coefficient of \( x^3 \)) = **343**
- Second blank (coefficient of \( x^2 \)) = **-294**
- Third blank (coefficient of \( x \)) = **84**
- Fourth blank (constant term) = **-8**

Would you like more details on any step?

### Related questions:
1. How is the binomial expansion formula derived?
2. How can you use Pascal's triangle to expand binomials?
3. What are some practical applications of binomial expansions?
4. Can the binomial theorem be applied to non-integer exponents?
5. How does the expansion differ for \( (7x + 2)^3 \)?

**Tip**: When expanding binomials, always check if you can simplify terms or factor common expressions after the expansion!

Learn how to expand the binomial function f(x) = (7x - 2)^3 using the binomial expansion theorem. This solution provides a step-by-step breakdown, from calculating each term to finalizing the expanded polynomial. Suitable for high school students studying algebra and binomial expansions.

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